Price Variation Limits
and Financial Market Bubbles:
Artificial Market Simulations
with Agents' Learning Process
SPARX Asset Management Co. Ltd.
Takanobu MIZUTA*
The University of Tokyo
Kiyoshi IZUMI The University of Tokyo
CREST PRESTO, JST
Shinobu YOSHIMURA University of Tokyo
* mizutata@gmail.com
* http://www.geocities.jp/mizuta_ta/
1
http://www.slideshare.net/mizutata/cifer2013
Thank you very much. I’m Takanobu MIZUTA from SPARX Asset Management.
I’m also belonging to The University of Tokyo.
Today, I’m going to give a presentation under the title of This.
1

Introduction
restrict trades market more
Price Variation Limit ？
efficient or not？
out of certain
price ranges to Prevent
Overshoot
Experiment 1 Experiment 2
Replicate Overshoot? Parameters’
Artificial Market Model Condition
Verified by ⊿Ppl, tpl
Hazard Rate (Agent Based Simulation)
NEED NEED
Learning Process
∆Ppl / t pl < v
Should Avoid
∆Ppl / t pl << v
2
http://www.slideshare.net/mizutata/cifer2013
■ Financial exchanges sometimes employ a “price variation limit”, which restrict
trades out of certain price ranges to avoid sudden large price fluctuations,
“overshoots” overshoots occur in bubble and crash in real financial market
■ There is a debate over whether the price variation limit makes financial market
more efficient or not.
■ To investigate price variation limits, we built an artificial market model
implementing a learning process.
■ Experiment 1, we will show that the model should be implemented the learning
process to replicate overshoot.
■ Experiment 2, We will also show that a parameters' condition of the price
variation limit to prevent overshoot.
2

restrict trades market more
Price Variation Limit ？
efficient or not？
out of certain
price ranges to Prevent
Overshoot
Experiment 1 Experiment 2
Replicate Overshoot? Parameters’
Artificial Market Model Condition
Verified by ⊿Ppl, tpl
Hazard Rate (Agent Based Simulation)
NEED
NEED
Learning Process
∆Ppl / t pl < v
Should Avoid
∆Ppl / t pl << v
3
http://www.slideshare.net/mizutata/cifer2013
First, I will describe our artificial market model.
3

Artificial Market Model (Agent Based Model)
Chiarella et. al. [2009]
● Continuous Double Auction
⇒ to implement realistic price variation limit
● Agent model is Simple
⇒ to avoid arbitrary result “Keep it short and simple”
heterogeneous 1000 agents
Expected Return
wi , j
1  P 
ret, j =  w1, j log ft + w2, j rht, j + w3, j ε tj 
 
∑i wi , j  P Strategy
Weight 
Fundamental Technical noise
↑ Different
+ Learning Process Our Original for each agent
↑ need to replicate an overshoot
Good Performance Strategy wi , j is up 4
Bad Performance Strategy wi , j is down
We built an artificial market model on basis of Chiarella et. al. [2009].
●Pricing mechanism is Continuous Double Auction
It is not simple, but, we need to implement realistic price variation limit
●Agent Model is Simple. This is to avoid arbitrary result, “Keep it short and
simple” principle.
We think Artificial Market Models should explain Stylized Facts as Simply as
possible.
There are heterogeneous 1000 agents. All agents calculate Expected Return
using this equation,
And, the strategy weights are different for each agent
・First term is a Fundamental Strategy: When the market price is smaller than the
fundamental price, an agent expects a positive return , and vice verse.
・Second term is a technical strategy: When historical return is positive, an agent
expects a positive return, and vice verse.
・Third term is noise.
Chiarella’s model did not include Learning Process, however,
We built Learning Process of agents, this is our Original.
We showed that learning process need to replicate an overshoot.
Agents are comparing Historical Return and Each Strategy’s Return.
・When the strategy’s return and Historical Return are Same Sign, Good
Performance Strategy, The strategy’s Weight is Up.
・When the strategy’s return and Historical Return are Opposite Sign, Bad
Performance Strategy, The strategy’s Weight is Down.
4

restrict trades market more
Price Variation Limit ？
efficient or not？
out of certain
price ranges to Prevent
Overshoot
Experiment 1 Experiment 2
Replicate Overshoot? Parameters’
Artificial Market Model Condition
Verified by ⊿Ppl, tpl
Hazard Rate (Agent Based Simulation)
NEED
NEED
Learning Process
∆Ppl / t pl < v
Should Avoid
∆Ppl / t pl << v
http://www.slideshare.net/mizutata/cifer2013 5
Next, I show simulation results of Experiment 1 about leaning process and
replicating overshoot
5

Experiment 1： Learning and Overshoot
Case 1
Fundamental Value
= constant
With
Learning Process
Case 2
Fundamental Value
× For
Without
→ rapid increment Learning Process
Each
(bubble trigger)
Cases
6
We examined two Cases, Case 1, Fundamental Value is constant,
Case 2, Fundamental value is rapid incremented like this. This is bubble inducing
trigger.
For Each cases, we examined With learning process And WithOut learning
process.
Therefore, we examined four cases in all.
6

Case 1： Fundamental Value = constant
case 1
10100
10050
10000
price
9950
9900
9850 non-learning learning
9800
0
100000
200000
300000
400000
500000
600000
700000
800000
900000
time
Small fluctuating around Fundamental Value 7
This Figure shows time evolution of market prices in case 1, Fundamental Value
is constant.
In both cases, With learning process and without learning process. the results are
very similar,
The prices were small fluctuating around Fundamental Value, here, Ten
Thousand
7

Case 2： Fundamental Value → rapid increment (bubble trigger)
case 2
20000
18000
16000
price
14000
non-learning
12000
learning
10000 fundamental value
8000
0
100000
200000
300000
400000
500000
600000
700000
800000
900000
time
Only with learning process, Overshoot occurred
8
This Figure shows time evolution of prices in case 2.
Fundamental value was changed at this time, increased to New Fundamental
Value, Fifteen Thousand.
This is the bubble inducing trigger.
Without Learning Process, Black line, Overshooting was not occurred.
On the other hand, with Learning Process, Red line,
the price went far beyond the new fundamental value.
Only with learning process, Overshoot occurred.
8

Traditional Stylized Facts
case 1 case 2
non-learning learning non-learning learning
kurtosis 3.018 5.394 2.079 3.180
lag
1 0.134 0.125 0.219 0.325
2 0.101 0.105 0.164 0.293
3 0.076 0.087 0.133 0.274
autocorrelation 4 0.060 0.074 0.118 0.261
coefficient for 5 0.052 0.061 0.108 0.253
square return 6 0.040 0.054 0.100 0.247
7 0.036 0.048 0.092 0.241
8 0.030 0.045 0.087 0.237
9 0.026 0.039 0.082 0.238
All cases replicated: Fat Tail and Volatility Clustering 9
This Table lists Traditional stylized facts in each case.
In all cases, both kurtosis and autocorrelation for square returns for all i are
positive.
This means that all cases replicate Traditional stylized facts: fat-tail and volatility-
clustering.
9

Hazard Rate (similar to “run test”)
New Stylized fact to verify model replicating overshoot
H(i) conditional probability that
sequence of positive return ends at i, given that it lasts until i.
For Example i=3、H(3)
Time ・・・・ １ ２ ３ ４
H(3)： Probability of
Return
4th Return Negative
Empirical Studies:
Any cases: H(i) < 50%,
Overshoot period: H(i) decline with I rapidly
McQueen and Thorley [1994], Chan et. al. [1998]
⇒ Overshoot returns tend to continue to be positive
this tendency stronger continuing positive returns longer
10
We propose Hazard Rate as New Stylized fact to verify model replicating
overshoot
Hazard Rate Hi is conditional probability that sequence of positive return ends at i,
given that it lasts until i.
For Example i=3, H3 means like this.
1st, positive return, 2nd, positive, 3rd positive,
In this condition, H3 is probability of 4th return become negative.
Empirical Studies showed that, Any cases, Hi for most of i are smaller than 50%
And when including overshoot period, Hi decline rapidly with i,
This show that the overshoot returns tend to continue to be positive
And this tendency stronger continuing positive returns longer
10

New Stylized Facts: Hazard Rate H(i)
case 1 case 2
non-learning learning non-learning learning
i
1 56% 55% 56% 55%
2 55% 52% 55% 50%
3 55% 50% 53% 45%
4 54% 49% 52% 40%
hazard rate 5 54% 45% 48% 36%
6 53% 44% 45% 29%
7 52% 41% 40% 26%
8 52% 40% 35% 22%
9 53% 40% 30% 19%
Only with Learning process → Verified by Hazard Rate
And Only Case 2 with Learning ⇒ Replicating Overshoot
11
This Table lists New Stylized Facts: Hazard Rate in each case.
In case 2 with learning, hazard rate declined rapidly.
This case can replicate a significant Overshoot like actual markets.
On the other hand, the case without learning, hazard rate dose not declined
rapidly.
The case can not replicate Overshoot.
Case 1, without learning, Hazard rates are upper 50% for all i.
This is Not consistent with empirical study.
On the other hand, Case 1, with learning, Hazard rates for most of i are smaller
than 50%, even when price fluctuations are stable.
This consistent with empirical study.
Therefore, only cases with Learning Process were verified by Hazard Rate,
and only Case 2 can replicate overshoot.
11

Result Summary Experiment 1： Learning and Overshoot
Not-Consistent with Consistent with
Empirical study Empirical study
↑ ↑
Without With
Learning Process Learning Process
Case1 Stable Stable
Fundamental Value Not-Verified by Verified by
= constant Hazard Rate Hazard Rate
Case2 No-Overshoot Overshoot
Fundamental Value Not-Verified by (Bubble & Crush)
→ rapid increment Hazard Rate Verified by
Hazard Rate
12
Result Summary Experiment 1 relationship between Learning process and
replicating Overshoot
The cases With learning process, both case 1 and case 2, were Consistent with
Empirical study verified by Hazard Rate.
And case 2 can replicate overshoot, bubble and crush
The cases Without Learning Process were Not consistent with Empirical study
Not verified by Hazard Rate.
12

restrict trades market more
Price Variation Limit ？
efficient or not？
out of certain
price ranges to Prevent
Overshoot
Experiment 1 Experiment 2
Replicate Overshoot? Parameters’
Artificial Market Model Condition
Verified by ⊿Ppl, tpl
Hazard Rate (Agent Based Simulation)
NEED
NEED
Learning Process
∆Ppl / t pl < v
Should Avoid
∆Ppl / t pl << v
13
http://www.slideshare.net/mizutata/cifer2013
Next, I show Simulation Results about Price Variation Limit.
13

Experiment ２： Price Variation Limit
Price Variation Limit
t −t p l
Can Not order Outside P ± ∆Ppl
Two Constant t pl Limit time Span
Parameters: ∆Ppl Limit price Range
changed to
Buy order above here
t − t pl
P + ∆Ppl
∆Ppl
t − t pl
t pl P
t − t pl
P ∆Ppl
Market Price
Pt changed to
t − t pl
Sell order under here
P − ∆Ppl
14
We modeled the price variation limit like this.
There are two constant parameters.
Tpl is a limit time span, and ⊿Ppl is limit price range.
We referred market price Before tpl, Pt-tpl.
and any agents can Not order OutSide from Pt-tpl - ⊿Ppl to Pt-tpl + ⊿Ppl
Concretely,Any buy order prices above here, they are changed to this price.
and any sell order prices under here they are changed to this price.
14

Case 2 & with Learning, Price Variation Limit
case 2, learning
20000
18000
16000
price
14000
12000 non Price Limit
Price Limit
10000 fundamental value
8000
0
100000
200000
300000
400000
500000
600000
700000
800000
900000
time
Overshooting was vanished in price variation limit
But, price took longer to reach new fundamental price 15
This Figure shows time evolution of prices in case 2 with learning,
comparing the case implemented price variation limit and not implemented.
Overshoot was vanished In the case implemented price variation limit
However, price took longer to reach new fundamental price.
In the case not implemented price variation limit, price took faster to reach new
fundamental price.
Implemented case, slower to reach new fundamental value, converging is slower.
15

Overshooting Amplitude and Converging Speed
case 2, learning
20000
reach time to new
18000 max price from
fundamental value new fundamental value
16000
price 14000
12000 non Price Limit
Price Limit
10000 fundamental value
8000
0
100000
200000
300000
400000
500000
600000
700000
800000
900000
time
Preventing Overshooting and Immediate Converging
→ no market achieves both at once
Optimization of t pl and ∆Ppl
16
Next, I investigated relationship between Overshooting Amplitude and
Converging Speed.
I measured, here, reach time to new fundamental value, and here, max price
from new fundamental value.
We want Preventing Overshooting and Immediate Converging.
However, No market achieves both at once.
Therefore, it is important to Optimize of these Two parameters.
16

Max Price from New Fundamental Value
max price from new tpl
fundamental value 1000 2000 3000 4000 7000 10000 15000 20000 25000 30000 40000 50000
100 1,374 475 285 205 115 76 --- --- --- --- --- ---
200 3,236 1,421 728 485 224 146 97 72 --- --- --- ---
300 3,450 3,020 1,378 900 387 233 134 106 90 86 --- ---
400 3,482 3,501 2,381 1,307 565 339 164 112 95 93 85 ---
700 3,347 3,470 3,366 3,311 1,395 734 363 211 120 94 90 83
⊿Ppl 1000 3,494
1500 3,512
3,357
3,356
3,229
3,424
3,578
3,404
2,633 1,388 681
3,407 3,019 1,384
421
607
267
286
122
208
93
184
91
183
2000 3,454 3,595 3,334 3,317 3,682 3,493 2,408 1,094 580 452 411 418
2500 3,475 3,357 3,497 3,368 3,262 3,466 3,524 2,007 1,442 508 103 100
3000 3,436 3,443 3,598 3,431 3,268 3,330 3,365 2,921 1,602 1,190 980 931
4000 3,359 3,398 3,374 3,607 3,673 3,378 3,338 3,351 3,433 1,701 477 401
5000 3,556 3,467 3,317 3,509 3,440 3,162 3,486 3,311 3,320 3,231 1,345 104
∆Ppl v: Converging Speed
Blue area: < v ≅ 0.128 without price variation limit
t pl
Preventing Overshooting
∆Ppl / t pl smaller (to upper right) ⇒ Overshoot smaller
NEED to prevent overshoots ∆Ppl / t pl < v 17
This table lists Max Price from New Fundamental Value in various Tpl and ⊿Ppl
Blue area, ⊿Ppl over Tpl is smaller than V
V is a converging speed without price variation limit, approximately 0.128.
As you see, in this area, max price small, this means preventing overshoot.
This is smaller, to upper right area, overshoot are smaller.
Therefore, we found that this is smaller than V is needed to prevent overshoot.
17

Reach time to New Fundamental Value
reach time to new
fundamental value
tpl
(x 1000) 1000 2000 3000 4000 7000 10000 15000 20000 25000 30000 40000 50000
100 55 104 157 216 382 555 --- --- --- --- --- ---
200 40 55 79 103 181 261 385 509 --- --- --- ---
300 39 41 55 70 120 171 257 342 425 506 --- ---
400 39 39 44 55 91 128 195 259 324 385 505 ---
700 40 39 40 39 55 75 113 148 185 229 302 371
⊿Ppl 1000 1500
39
39
39
39
39
39
39
39
41
39
54
40
78
54
104
72
127
93
150
113
193
146
234
175
2000 39 39 40 40 39 39 43 56 71 85 106 126
2500 39 39 39 39 39 39 39 46 53 60 74 86
3000 39 39 39 39 40 39 39 40 48 56 70 79
4000 39 39 39 39 39 39 40 39 39 44 63 76
5000 39 39 39 39 39 40 39 39 40 40 40 47
∆Ppl
Blue area: < v ≅ 0.128 v: Converging Speed
t pl without price variation limit
∆Ppl / t pl smaller (to upper right) ⇒ converging speed slower
Should Avoid Not to be converging slower ∆Ppl / t pl << v 18
This table lists Reach time to New Fundamental Value in various Tpl and ⊿Ppl
Blue area, this is smaller than V.
This is smaller, to upper right area, converging speed are slower.
Therefore, we found that it this is too smaller than V is should be Avoid Not to be
converging slower.
18

Result Summary Experiment 2: about Price Variation Limit
Price Variation Limit
prevent Overshooting and Converging Slower
Optimization of t pl and ∆Ppl
∆Ppl / t pl smaller ⇒ Overshooting smaller
NEED to prevent overshoots
∆Ppl / t pl < v ∆Ppl / t pl << v
Should Avoid Not to be converging slower
∆Ppl / t pl smaller ⇒ converging speed slower
v: Converging Speed
without price variation limit 19
Result Summary Experiment 2 about Price Variation Limit.
Price Variation Limit prevents Overshoot and cause Converging speed Slower to
new fundamental value
Therefore, it is important to Optimize of these two parameters.
When this is smaller, overshooting smaller.
We found that it needs smaller than, V to prevent overshoots
On the other hand, this is smaller, converging speed is slower.
We found that it should be avoid too smaller than V Not to be converging slower
19

Summary
restrict trades market more
Price Variation Limit ？
efficient or not？
out of certain
price ranges to Prevent
Overshoot
Experiment 1 Experiment 2
Replicate Overshot? Parameters’
Artificial Market Model Condition
Verified by ⊿Ppl, tpl
Hazard Rate (Agent Based Simulation)
NEED NEED
Learning Process
∆Ppl / t pl < v
Should Avoid
∆Ppl / t pl << v
20
http://www.slideshare.net/mizutata/cifer2013
I summarize this presentation
■ To investigate price variation limits, we built an artificial market model
implementing a learning process.
■ Experiment 1, we showed that the model should be implemented the learning
process to replicate overshoot.
■ Experiment 2, We also showed that a parameters' condition of the price
variation limit to prevent overshoot.
20

That’s all for my presentation.
Thank you very much
for your cooperation !
21
http://www.slideshare.net/mizutata/cifer2013
Could you say that again? (もう一度、おっしゃっていただけますか？)
I don’t quite understand your question. (ご質問の趣旨が良く分からないのですが)
Could you please rephrase your question? (ご質問を分かりやすく言い換えていただ
けますか)
So, you are asking me about.... (つまり、お尋ねの内容は...ですね)
I totally agree with you. (私も全くあなたと同意見です)
That’s a very challenging question for me to answer. (それは私にとって非常に答え
がいのある質問です)
That’s a question I’m not sure I can answer right now. (そのご質問にすぐお答えで
きるかどうか分かりません)
It would require further research. (さらなる研究結果を待ちたい)
You are right on that point. (その点に関してはあなたが正しい)
Our method will not solve the problem. (我々の方法ではその問題は解決できない)
21

Appendix
22
22

Artificial Market Model (Agent Based Model)
On basis of Chiarella et. al. [2009]
+ Learning Process of agents
comparing between the case with Learning Process and without
Feature of our model
○ agent model is Simple
→ to avoid arbitrary result “Keep it short and simple”
Models should explain stylized facts as simply as possible
● pricing mechanism is Continuous Double Auction
→ not simple to implement realistic price variation limit
☆ Learning process
→ agents switch strategy, fundamental or technical
An overshoot occurred in the case with the learning process
but did not occur in the case without the process
23
We built an artificial market model on basis of Chiarella et. al. [2009].
Chiarella’s model did not include Learning Process, however,
We built Learning Process of agents.
And we are comparing between the case with Learning Process and without it.
Our Agent Model is Simple. This is to avoid arbitrary result, “Keep it short and
simple” principle.
We think Artificial Market Models should explain Stylized Facts as Simply as
possible,
Our pricing mechanism is Continuous Double Auction
It is not simple, but, we need to implement realistic price variation limit
Learning process
Here, Learning process means agents are switching strategy, fundamental
strategy or technical strategy.
We will show that, an Overshoot occurred in the case With the learning process,
however, overshoot did not occur in the case WithOut the process
23

Agent Model
j: agent number (1000 agents) Historical Return
t −τ j
ordering in number order rht, j = log( P t / P )
t: tick time
Technical
Expected Return
1  P 
ret, j =  w1, j log ft + w2, j rht, j + w3, j ε tj 
 
∑i wi , j  P 
Parameters for agents Fundamental noise
wi , j
and τj Pf Fundamental Price
Random of
ε tj
10000 = constant
Random of Normal
Uniform Distribution
P t Market Price at t Distribution
Average=0
i=1,3: 0～1 σ=3%
wi , j Expected Price
i=2: 0～10
τj 0～10000 Pet, j = P t exp(ret, j ) 24
Next, I will describe agent model.
All agents calculate Expected Return using this equation.
First term is a Fundamental Strategy:
When the market price is smaller than the fundamental price, an agent expects a
positive return , and vice verse.
Second term is a technical strategy:
When historical return is positive, an agent expects a positive return, and vice
verse.
Third term is noise,
After the expected return has been determined, an expected price is determined
like this.
And, agents order base on this Expected Price.
24

Learning Process
Expected Return
1  P 
ret, j =  w1, j log ft + w2, j rht, j + w3, j ε tj 
 
∑i wi , j  P 
Historical Return Compare each Strategy
t −t l
rl = log P / P
t t
Same Sign Opposite Sign
good performer Strategy bad performer Strategy
wi , j ← wi , j + krlt [0, ( wi , max − wi , j )] wi , j ← wi , j − krlt [0, wi , j ]
Weight Up Weight Down
With 1% probability： [a,b]: Random Uniform Distribution
wi , j ← [0, wi ,max ]
Reset from a to b
25
We also developed a model implementing a learning process of agents.
Agents are comparing Historical Return and each Strategy’ term, Fundamental
strategy term, and Technical strategy term.
When the strategy’s expected return and Historical Return are Same Sign,
This means good performer Strategy.
The strategy’s Weight is Up.
When the strategy’s expected return and Historical Return are Opposite Sign,
This means bad performer Strategy.
The strategy’s Weight is Down.
We also added random learning.
In this way, agents learn better parameters and switch to the investment strategy
that estimates correctly.
25

Order Price and Buy or Sell
price
Pet, j + Pd
P t Sell （one unit） Pot, j > Pet, j
Order Price Random Expected Price
t t
P Uniform
o , j （about±10%） P e, j
Buy (one unit) P t < P t
o, j e, j
Pet, j − Pd
To Stabilize simulation for continuous double mechanism,
Order Prices must be covered widely in Order Book.
26
Next, agents determine order price and, buy or sell.
To Stabilize simulation runs for the continuous double mechanism,
Order Prices must be covered widely in Order Book.
We modeled an Order Price, Po, by Random variables of Uniformly distributed in
the interval from Expected Price, Pe, minus constant, Pd, to Pe plus Pd.
And then,
When Po lager than Pe, the agent orders to sell one unit.
When Po smaller than Pe, the agent orders to buy one unit.
26

Hazard Rate
case 2 lerning
non Price Limit Price Limit
i
1 55% 55%
2 50% 53%
3 45% 49%
4 40% 47%
hazard rate 5 36% 44%
6 29% 42%
7 26% 40%
8 22% 36%
9 19% 31%
Hazard Rates increased
→ Result by preventing bubble
27
Hazard Rate
Hazard Rates increased in implemented price variation limit
This Result shows preventing bubble
27

Strategies Weight
case 2, learning
10% 100%
9% 98%
8% 96%
fundamental weight
technical weight
7% 94%
6% 92%
5% 90%
4% 88%
3% 86%
2% fundamental weight (left axis) 84%
1% technical weight (right axis) 82%
0% 80%
0
100000
200000
300000
400000
500000
600000
700000
800000
900000
time
During overshooting, switching fundamental to technical
Consistent with empirical studies
28
[Frankel and Froot, 1990], [Yamamoto and Hirata, 2012]
I also show Strategies weight, in the case 2 with learning
which case include overshoot.
During overshooting, agents are switching fundamental strategy to technical
strategy.
This is consistent with empirical studies, like these studies.
28

Strategies Weight
case 2, learning, Price Limit
10% 100%
9%
8% 95%
fundamental weight
technical weight
7%
6% 90%
5%
4% 85%
3%
fundamental weight (left axis)
2% 80%
1% technical weight (right axis)
0% 75%
0
100000
200000
300000
400000
500000
600000
700000
800000
900000
time
Agents Not switching to the technical strategy
→ prevented overshooting. 29
I also show Strategies weight, in the case 2 with learning, implemented price
variation limit
Agents are not switching fundamental strategy to technical strategy.
This causes preventing overshooting, bubble and crash.
29